The overall aim of this project is to better understand biological evolution, speciation, and diversification through a study of the dynamic (and static) behavior of various mathematical models describing populations, species and clades. Using a combination of analytical methods and extensive numerical simulations, a number of mathematical models of evolutionary diversification taking into account various factors operating in natural populations will be studied. Both standard population genetics approaches and novel methods recently developed in theoretical evolutionary biology, mathematics, and physics will be used. This project has 3 interrelated specific aims. 1. Build mathematical foundations of a general theory of speciation, develop a mathematical theory of adaptive radiations, and formulate and study mathematical models linking evolutionary processes at different spatial and temporal scales (from speciation to adaptive radiation to macro evolutionary patters). 2. Establish closer connections between theory and empirical data through a better understanding of the implications of theoretical results for speciation in specific groups of organisms, exploration of how dynamical models of speciation can guide the development of statistical methods and hypotheses utilizing emerging comparative genomic data, and examination of how the data on molecular phylogenies and spatial range distributions can be used to infer the history of speciation and clade diversification. 3. Study the coevolutionary dynamics using explicit genetic and spatial models to analyze how antagonistic and mutualistic within- and between-species interactions depending on many loci and traits affect the evolutionary dynamics of populations in panmictic and spatially-structured systems. The studies proposed here may improve our understanding of the mechanisms responsible for the origin and maintenance of biodiversity and may be important for understanding and controlling the dynamics of host-pathogen interactions affecting human health.